Weak Interactions. Feynman Rules for the Muon Decay Fermi s Effective Theory of the Weak Interaction. Slides from Sobie and Blokland

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1 Wak ntractions Fynan uls for th Muon Dcay Fri s Effctiv Thory of th Wak ntraction osons lids fro obi and lokland Physics 424 Lctur 20 Pag 1

2 ntrdiat ctor osons Lik ED and CD, th wak intraction is diatd by spin-1 (vctor) particl xchang. Unlik th photon and gluons, th wak diators ar assiv: This ans that th longitudinal polarization od is availabl, for a total of 3 indpndnt polarizations. Physics 424 Lctur 20 Pag 2

3 Propagator for th and osons W nd to gnraliz th photon propagator in ordr to account for th ass, bosons., of th intrdiat vctor This probl is or subtl than it looks. W will us th so-calld unitary gaug propagator: ( "$% '& "$#! Physics 424 Lctur 20 Pag 3

4 1 0 ) 0 ) Chargd Currnt rtx Th bosons diat chargd currnt (CC) wak intractions. Thy coupl to lptons via 34 *,+.- W will look at wak intractions involving quarks soon. Notic that this intraction ixs vctor and axial vctor trs. W call this a intraction and it lads to parity violation. Physics 424 Lctur 20 Pag 4

5 ; : 3 i W J Z M C Y W A P Y W P M ` t p f 79< 798 Exapl: 8 = = 3@? = < >= ` % 'cb a ` # _ UH C.M.N JL j\j^k [\[^] T G DFEHG C hg f d UH C.M$N J T _ G D EHG l a ' b 'on U T rsehg t U G T G rsehg hg N q Physics 424 Lctur 20 Pag 5

6 ; : M t A f T M r t } w r } l 7u< 7 8 : UH T rsehg t UH G T G rsehg vg N q U U xzy T _ T hg f N q t U T _ U T (lctrons hav 2 spins and th nutrino has 1 Not th lading factor of spin). factors togthr first: U T To valuat th tracs, it hlps to bring th U T U T U T _ U T _ U U T T _ U T _ W do th sa thing for th first trac, wrapping around so as to avoid th tr. Physics 424 Lctur 20 Pag 6

7 ; : M t _ r } r } l ˆ M Œ f g _ _ Ž l g Œ Œ Š _ Š 7 < 7 8 : U x y T g f N q t U _ T -dpndnt trs do not contribut to ths tracs, so w hav Th,Œ _ N x y CoŠ _ M CoŠ M M N f For th last stp, rbr that:,œ _ Physics 424 Lctur 20 Pag 7

8 ; : g w f g 7u< 7 8 : M N f xzy n th CM fra and nglcting th ass of th lctron, t r t r t r t r t r š Y D T Y D T q M N x y Physics 424 Lctur 20 Pag 8

9 ; : w f g W can quickly convrt this to a cross sction using Fri s Goldn ul: W T M N Ÿ š T M N œ 7u< 7u8 : Y D T q M N x y x y y y> JT y y ž œ qoÿ t taks a bit of ffort to show that Y D T y y y y but fro this point on, th calculation is trivial: Y D T ž œ hg f Y D T vg f qoÿ Physics 424 Lctur 20 Pag 9

10 : ; 3 d M t A f _ g 7 < 7 8 Muon Dcay 3? = = < = 8 =, th aplitud is hg f Again, working in th liit of U T rsehg t U G T rsehg g N q p, This is idntical to th aplitud in th prvious xapl, xcpt for but sinc ithr spinor givs us in th trac (sinc ), w can rcycl th rsult G M N f xzy Physics 424 Lctur 20 Pag 10

11 g «ª r w f g Prliinary inatics M N f x y scattring, w inc th kinatics of dcay ar diffrnt fro thos of will nd to start our work hr., and n th uon rst fra, t r t r t M N xzy Physics 424 Lctur 20 Pag 11

12 W J W J W J l Ÿ T inc xzy (via Fri s Goldn ul ) dpnds nontrivially on th dcay rat fro scratch with Fri s Goldn ul: x y, w will hav to work out Ÿ Ÿ Ÿ Ÿ As always, w gathr th factors of 2 and -function to do on of th intgrals. W ll do on, and w us th spatial parts of th first, sinc x y dpnds and w ll want our pnultiat rsult to dpnd on th lctron nrgy x y U Ÿ Physics 424 Lctur 20 Pag 12

13 ž y y «w Ÿ š Nxt, w will intgrat ovr Th Nxt ntgral. Working in sphrical coordinats, ³ ±o² y y whr is th angl btwn particls 2 and 4 ( E and ) x y ±o² whr rprsnts µ. -intgral, or w can sarch for a iraculous chang of variabls to sav th day. W could ithr us th chain rul to valuat th Physics 424 Lctur 20 Pag 13

14 w º ¹ T Th Miraculous Chang of ariabls t s staring right at us: µ ±o². W hav to b a littl bit carful hr, bcaus as Lt s rwrit ±o² ¹» Physics 424 Lctur 20 Pag 14

15 ½ ¼ y ½ º ¹ T T y ¾ ¾ ¾ Ä Â ¾ ¾ Consrvation of Enrgy: Algbraic, w hav µ Fro y -intgral is As a rsult, th ¹» providd that ¾ y y y oã ÀÁ Physics 424 Lctur 20 Pag 15

16 ¾ ¾ Â ÀÁ ¾ w Consrvation of Enrgy: ntuitiv Ä oã Í Æ>ÆÉÎ Æ>ÆËÏ ¾ ÅÇÆ>ÆÉÈ Æ>ÆËÊ Ì inc all thr final-stat particls ar assud to b asslss, nrgy and (3-) ontu ar th sa. Th su of th onta for th 3 final-stat particls ust b zro, thrfor no singl particl can hav or than half of th availabl nrgy and no two particls can hav lss than half of th availabl nrgy. Physics 424 Lctur 20 Pag 16

17 Ÿ Ÿ ntgral olving th turning to th dcay rat, x y ±o² š x y š to runs fro, thn rangs fro 0 to whr, if and x y (i.., work fro th outsid in). Now w can substitut for intgral. valuat th Physics 424 Lctur 20 Pag 17

18 Ÿ W J Ÿ š W W J Ô Ÿ š W W J Ÿ š Ô Ÿ š W š W J Ÿ Elctron nrgy spctru x y š Ñ Ð # M.N f g Ò$Ó Ñ» Ð # J M$N f g in sphrical coordinats and intgrating ovr th angls, w hav Writing J M$N f g J T M$N š f g Ô This dscribs th lctron-nrgy spctru of uon dcay. Physics 424 Lctur 20 Pag 18

19 W š Õ J T Ö Ô Physics 424 Lctur 20 Pag 19

20 W š Ö Õ Elctron nrgy spctru data J T Ô Physics 424 Lctur 20 Pag 20

21 Û à Ü Ü á Û Þ Ü Ü Û 1 Ü Þ Th Muon Dcay at ntgrating ovr th lctron nrgy, w (finally) obtain th uon dcay rat: â ß # ÙØ â ã â Þ Ý Ú Ø Þ Ý ä Ø ä Þ Ý This is xactly as w prdictd! (Up to th inor costic factor of.) å Ý Physics 424 Lctur 20 Pag 21

22 æ Þ Ý ê 1 Ü Ü ê ) Th Fri Coupling Constant n th liit of ratio of and Ø, our rsults always dpnd on th, and not th two constants sparatly. Dfin th Fri coupling constant, çlè, by é Ø ç è This allows us to writ th uon lifti as ç è Using quation: and, w actually dtrin ç è fro this 1 ) ìë ç è Physics 424 Lctur 20 Pag 22

23 ) íî Ø å â How Wak is th Wak ntraction? With th uon lifti asurnt giving us é 1 ) ë Ø ç è w can us th ass asurnt to dtrin Ø. Th rsult, Ø Ý Ø indicats that th wak intraction is inhrntly strongr than th lctroagntic intraction! t is only th supprssion factor which aks th wak forc s so fbl. Physics 424 Lctur 20 Pag 23

24 ð «ï ñ «Nutron Dcay nsofar that nutron substructur dosn t co into play, w could odl nutron dcay as a wak intraction procss uch lik uon dcay: E š ð Ô ð Toð n uon dcay, all 3 final-stat particls (, E, and ) ar ssntially asslss. Physics 424 Lctur 20 Pag 24

25 ó T TÔ inatics of Nutron Dcay n nutron dcay, th proton ass is obviously quit larg. n addition, th ass of th lctron (0.5 M) is a significant fraction of th nutron-proton ass diffrnc (1.3 M), so w cannot ignor. As a rsult, th phas-spac calculation for nutron dcay is or difficult than that of uon dcay. Consult Griffiths if you would lik to s th dtails. Using a pur ò vrtx factor, w obtain a nutron lifti of ôöõ Th xprintally asurd valu is 886 s (about 15 in). Physics 424 Lctur 20 Pag 25

26 ø ø 1 ) 0 0 ùý Effcts of ubstructur W should gnraliz th vrtx to ùý úùüû *,+ - Exprints indicat that ùüû CC: Consrvd ctor Currnt PCAC: Partially Consrvd Axial Currnt Physics 424 Lctur 20 Pag 26

27 á 0 ÿ 0 Ý ã þ þ ) Ý 2 þ ã Pion Dcay Whil th dcays to via an lctroagntic intraction, th chargd pions ( and ) dcay to a lpton-nutrino pair through th wak intraction. n so rspcts, procss: ã ã þ úþ ã dcay can b rgardd as a scattring 34 Although w know how th coupls to quarks (and w will look at this soon), w would vntually nd to know to calculat. ê Physics 424 Lctur 20 Pag 27

28 ) Ý 2 ) Ý Ý An Altrnativ Approach f w r going to hav so unknown factor apparing in our rsults, w should ak th rst of th calculation as sipl as possibl. Lt s odl dcay by: 34 whr th vrtx factor intraction at th blob is dscribd by th Ø ø é Physics 424 Lctur 20 Pag 28

29 ï Ÿ t f M N q A Th Pion Dcay Aplitud With our ansatz for th intraction, th aplitud is U T rsehg hg inc th pion is a spin-0 particl,. Th only scalar w can ak fro can only dpnd on th pion ontu, is, so is, in fact, constant! W call th pion dcay constant, and xprints suggst that Ô Physics 424 Lctur 20 Pag 29

30 M t N A M N _ w r } l M r } _ N Š U _ M _ r q _ N w Th Pion Dcay Aplitud U T rsehg hg f q x y hg f q t U T _ U T t _ U T g f q. Th trs will not contribut, as thy all involv tracs of an odd # of -atrics. Finally, th -tnsor producd by th trac involving will vanish whn contractd with. U U T T W start by using t _ M x y _ hg f q Physics 424 Lctur 20 Pag 30

31 M _ r q _ N r M N T q w M N w Th Pion Dcay Aplitud t _ M x y _ hg f q t g f : W can valuat th various dot products by using iilarly, T T xzy hg f Physics 424 Lctur 20 Pag 31

32 W J Th Pion Dcay at Fro Fri s Goldn ul, yy y qoÿ qualling th nutrino nrgy, y y With y y M.N w f g š Ÿ Physics 424 Lctur 20 Pag 32

33 W J š» Ÿ»«p» T» ª p Th Pion Dcay at MoN f g Th othrwis unknown can b xtractd fro this xprssion. ttr yt, lt s copar th cancl : dcay rats to lctrons and uons so as to E» Ÿ E Ÿ» urprisingly, th uon od is havily favord in spit of th sallr phas spac availabl. Th supprssion of th lctron od can b undrstood in trs of angular ontu. Physics 424 Lctur 20 Pag 33

34 3 þ ã What About uarks? For lptons, th coupls within a particular gnration: 3 ê 8 3? < Things ar or coplicatd for quarks, as th can ix gnrations: couplings ù Physics 424 Lctur 20 Pag 34

35 ã ) 0 þ ) 0 þ W J W J ñ C Cabibbo angl 1 ù 0 *, *,+ - M$N» p» Ÿ f g š Ÿ M.N "!!» p» f g š Ÿ Physics 424 Lctur 20 Pag 35

36 ñ ( ( T l š T T l &» ª p» ª p T T T Ô & aon and pion dcays Th ratio of th widths!!» p» #%$ p»» Ÿ sconds,» and» '& Th liftis of th pion and kaon ar rspctivly. is 64%.» is 100% and» Ÿ Th branching ratios for M. T M and Ô W nd othr asurnts to gt š) & dgrs and sult is Ô & ) * & Hnc Physics 424 Lctur 20 Pag 36

37 + uary of chargd-wak intraction Th bosons diat chargd wak intractions and th Fynan ruls incorporat th ass of th and th parity-violating natur of th wak intraction. With ths Fynan ruls, w ar abl to calculat th lifti of th uon. At low nrgis, th rlvant wak intraction paratr is th Fri coupling constant. çlè Th wak intraction is not wak bcaus th coupling constant is sall, but rathr bcaus of th larg ass of th virtual which ust b xchangd. Th boson is rsponsibl for both nutron dcay and th dcay of th chargd pion. Physics 424 Lctur 20 Pag 37

Andre Schneider P621

Andre Schneider P621 ndr Schnidr P61 Probl St #03 Novbr 6, 009 1 Srdnicki 10.3 Vrtx for L 1 = gχϕ ϕ. Th vrtx factor is ig. ϕ ig χ ϕ igur 1: ynan diagra for L 1 = gχϕ ϕ. Srdnicki 11.1 a) Dcay rat for th raction ig igur : ynan